An empty multistorey parking garage under construction in Mexicali collapsed and broken gas pipes sparked a number of fires, Eduardo Sandoval, a Baja California civil protection official, told Mexican radio.
'Scene of mild panic'
Al Jazeera's Mike Kirsch reporting from Mexicali said it was too early to properly assess the situation following the quake.
"While there were a few hours of daylight to assess the damage, emergency officials were not able to cover the entire city to find out how much damage was done or if there are more casualties," Kirsch said.
"There is a scene of mild panic on the Mexicali side of the border. Many motorists have been prevented from coming into the US.
"We are hearing reports of one woman who was escorted across the border from Mexicali after giving birth in a powerless hospital.
"This city of a million people is in comlpete darkness."
The quake also shook high-rise buildings as far away as Los Angeles and Las Vegas in the US, hundreds of kilometres away.
Our correspondent said tremors lasting between 30 seconds and a minute were felt in the US city of San Diego.
Some of its neighborhoods reported minor structural damage and burst water pipes and callers to local radio said the rolling tremor made it hard to keep vehicles on the road.
Miguel Almaguer, a reporter for NBC television in Calexico, California, told Al Jazeera that strict safety standards had meant that the number of deaths was low.
"Here in California there are strict building codes that even older buildings have to comply with," he said.
"So while we have seen buildings that have been damaged we have not seen toppled buildings. And that has helped in avoiding a big death toll."
Sunday's relatively shallow quake was initially reported to be 6.9 in magnitude but the US Geological Survey later revised it to 7.2 - one of the strongest to hit the area in decades.
Southern California is prone to frequent quakes, and many residents fearfully anticipate the next big one.
The last to cause major damage was the 6.7 magnitude Northridge quake in 1994 that left 57 dead, 9,000 injured and resulted in about $40bn in property damage.
Three aftershocks of magnitudes 5.1, 4.5 and 4.3 followed Sunday's quake in an area where minor tremors of 3.0-magnitude had been occurring all week.